Two congruent isosceles triangles and three lines. Prove triangle AEF is also isosceles.

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## Solution

Triangles ABD and CBE are similar since they are both isosceles and they have the same top angle at vertex B. Therefore also ∠EAF equals ∠AEF and hence △AEF is isosceles.

An interesting finding by Ignacio Larrosa Cañestro is that the locus of point F seems to be a hyperbola.

## Poem

The fallen pine

the tree is sad

he is just be cut

by human hands

like a giant

his crown to earth

he is broken and dirty

no more rain

no more life

no more thunder

just a yearning

just a burning

a windswept pine tree

when the pine tree fell

just a big sadness

## 2 replies on “The fallen pine”

I’m not totally sure that the locus is a hyperbola … It seems, but …

In the picture below I show the formula for F if AB is fixed and h is variable.

The graph is no hyperbola.